Wikipedia article:

Map showing all locations mentioned on Wikipedia article:

Normally an abbot is the superior of an abbey (a monastery), and
exercises authority over a religious family of monks. His authority extends only as far as the
monastery's walls, or only to the monks who have taken their
vows in his monastery. A
territorial abbot - also called an abbot
nullius diœceseos, Latin
"belonging to no diocese"), or
abbreviated abbot nullius, on the other hand,
functions additionally as the ecclesiastical governor for a
territory around the monastery, in much the same way a bishop does for a diocese.

The practice arose in part because abbeys have sometimes served as
missions. A monastery was
sometimes erected in territories where Christianity was first being preached, or in
remote and poor areas or areas still being settled. As the
monastery was the only ecclesiastical presence in this area, and as
the monks sometimes served as the parishclergy in church near the monastery, with even the
monastery's own chapel being a worship space for the laity who had settled nearby, the abbot of the
monastery, though having received only the priesthood in the sacrament of Holy
Orders, was invested with the same administrative authority
under canon law as a
diocesan bishop for a given territory around the abbey. Thus, with
the exception of actually ordaining new priests himself, the abbot
so empowered could do almost everything else a diocesan bishop
would for those under his care, including incardinate (that is, enroll
under his jurisdiction) even non-monastic priests and deacons for service in parishes.

Territorial abbeys still exist in some parts of the world: in
sparsely-settled or missionary areas, and in Europe where some
ancient abbeys nullius still retain their rights.

Though territorial (like other) abbots are elected by the monks of
their abbey, a territorial abbot can only receive the abbatial
blessing and be installed under mandate from the pope, just as a bishop cannot be ordained and installed
as ordinary of a diocese without such a mandate.

After the Second Vatican
Council, more emphasis has been placed on the unique nature of
the episcopacy and on the traditional organization of the church
into dioceses under bishops. As such, abbeys nullius have been
phased out in favor of the erection of new dioceses or the
absorption of the territory into an existing diocese. A few ancient
abbeys nullius still exist in Europe, and one in Korea.

List of territorial abbacies

There are only 11 remaining territorial abbeys (nearly all Latin rite, in fact titular sees) that have kept their title as bows
to their history. Most are in Europe:

Ecclesiastically united with South Korea, Tŏkwon had been vacant for over fifty years until
2005, when Fr. Francis Ri was appointed as the abbot. It has
not been united with any diocese in either South Korea or North
Korea due to the effective vacancy of the ones in North Korea and
the lack of effective jurisdiction applied by the South Korean
church.

Cluny (in
Burgundy; now united with the Diocese of Autun is the only
one in France. Historically Cluny was the mother house of
the Congregation of Cluny as a
result of the Cluniac monastic
reform of the 11th century, primarily in that it removed many
Benedictine abbeys under its
jurisdiction from local feudal allegiances (hence establishing
their independence) and had new ones founded. It became extremely
rich and influential within and beyond the Church.

Historically there have been more, such as

St.
Peter-Muenster , which from 1921 until 1998 served a remote area of
Saskatchewan, Canada (the abbey
still exists, but its territorial jusrisdiction and duties were
absorbed by the Diocese of Saskatoon).

Belmont Abbey-Mary, Help of
Christians, which was territorial abbey of half of North Carolina
from 1910 until 1960, when it lost its last extra-abbatial
territory. It was formally suppressed as a territorial abbey in
1977.